Assertion (A): If an iron rod is inserted into a steady current carrying solenoid, the current in solenoid decreases.
Reason (R): Magnetic flux linked with solenoid increases.
Inserting an iron rod increases magnetic flux \( \Phi \). By Lenz's law, this induces an opposing \( \text{EMF} \), causing current \( \text{I} \) to decrease during insertion.
Assertion (A): ac current flows through a bulb and a solenoid connected in series. If a soft iron core is inserted in the solenoid, the bulb glows much brighter.
Reason (R): The inductance of solenoid decreases on inserting soft iron core in it.
Inserting a soft iron core increases solenoid inductance \(L\), thus increasing inductive reactance \(X_L = omega L\). This increases circuit impedance \(Z\), reducing current \(I = V/Z\) and making the bulb dimmer. Both Assertion (A) and Reason (R) are false.
Assertion (A): A choke coil has the characteristic of high inductance and low resistance.
Reason (R): More is the inductive property of the choke coil, Power factor of the circuit approaches maximum.
A choke coil has high inductance and low resistance (A is true). Power factor is \(cos\phi = R/Z = R/sqrt{R^2 + X_L^2}\). Higher inductive property (large \(X_L\)) makes \(cos\phi\) approach minimum (0), not maximum. So R is false.
Assertion (A): In a series \(LCR\) circuit at resonance, the voltage across the capacitor or inductor may be more than the applied voltage.
Reason (R): At resonance in a series \(LCR\) circuit, the voltages across inductor and capacitor are out of phase.
At resonance, \(V_L = V_C\) but they are \(180^\circ\) out of phase. The applied voltage is \(V = IR\). Due to voltage magnification (high \(Q\) factor), \(V_L\) or \(V_C\) can be much greater than \(V\). Reason is true, but it doesn't explain *why* they can be larger than applied voltage, it explains why they cancel out to make \(V=IR\).
Assertion (A): Average power consumed in an \(AC\) circuit is equal to average power consumed by resistors in the circuit.
Reason (R): Average power consumed by capacitor and inductor is zero.
Average power in \(AC\) is \(P_{avg} = V_{rms} I_{rms} \cos\phi\). For pure inductor or capacitor, \(\phi = \pm \pi/2\) so \(cos\phi = 0\). Only resistors dissipate average power, \(P_{avg} = I_{rms}^2 R\). Hence, R correctly explains A.
Assertion (A): Peak voltage across the resistance can be greater than the peak voltage of the source in a series \(LCR\) circuit.
Reason (R): Peak voltage across the inductor can be greater than the peak voltage of the source in an series \(LCR\) circuit.
Peak voltage across resistor is \(V_R = I_0 R\). Peak source voltage is \(V_0 = I_0 Z\). Since \(Z \ge R\), \(V_R \le V_0\). So A is false. Peak voltage across inductor is \(V_L = I_0 X_L\). At resonance, if \(X_L > R\), then \(V_L\) can be greater than \(V_0\) (voltage magnification). So R is true. Thus, A is false and R is true.
Assertion (A): The power rating of an element in \(AC\) circuit refers to average power rating.
Reason (R): A given value for \(AC\) voltage or current is usually its average value.
Power rating of \(AC\) devices always refers to average power. However, \(AC\) voltage or current values (e.g., 220V) are typically Root Mean Square (RMS) values, not average values. For a full cycle, the average value of sinusoidal \(AC\) is zero.
Assertion (A): Average power consumed in a circuit is never negative.
Reason (R): Instantaneous power is always positive.
Average power consumed by passive circuits is non-negative. Instantaneous power \(p = vi\) can be negative during parts of an \(AC\) cycle, especially in reactive circuits, when energy is temporarily returned to the source.
Assertion (A): At an airport, a person is made to walk through the doorway of a metal detector.
Reason (R): Metal detector works on the principle of resonance in \(AC\) circuits.
Metal detectors use \(LC\) resonant circuits. When a metal object enters the coil's magnetic field, it changes the coil's inductance, altering the circuit's resonant frequency and triggering detection.
Assertion (A): Smaller the band width, sharper the resonance and easier it is to tune an \(LCR\) circuit.
Reason (R): Resonant frequency is arithmetic mean of half power frequencies.
Smaller bandwidth implies a higher quality factor \(Q\) and sharper resonance, leading to better frequency selectivity (easier tuning). Resonant frequency \(\omega_0\) is the *geometric mean* \(sqrt{\omega_1 \omega_2}\,\) not the arithmetic mean of half-power frequencies.